DOCTORAL THESIS
“Essays on Sticky Prices, Aggregate
Investment and Monetary Policy”
Lutz Weinke
Universitat Pompeu Fabra
Department of Economics and Business
Advisor: Jordi Gal´ı
Contents
Acknowledgements
iii
Foreword
iv
1 Pitfalls in the Modeling of Forward-Looking Price Setting
and Investment Decisions 1
1.1 Introduction . . . 1
1.2 The Model . . . 4
1.2.1 Households . . . 4
1.2.2 Firms . . . 6
1.2.3 A Short Note on Woodford’s Conceptual Mistake . . . 10
1.2.4 Market Clearing . . . 11
1.2.5 Linearized Equilibrium Conditions . . . 12
1.3 Equilibrium Dynamics . . . 15
1.3.1 Calibration . . . 15
1.3.2 Results . . . 16
1.4 Conclusion . . . 19
Appendix: Inflation Dynamics . . . 20
References . . . 21
2 New Perspectives on Capital and Sticky Prices 24 2.1 Introduction . . . 24
2.2 Theoretical Framework . . . 27
2.2.1 Firm-Specific Capital . . . 27
2.2.2 Rental Market for Capital . . . 28
2.3 Results . . . 30
2.4 Conclusion . . . 34
3 Firm-Specific Investment, Sticky Prices and the Taylor
Prin-ciple 37
3.1 Introduction . . . 37
3.2 The Model . . . 40
3.3 Results . . . 41
3.3.1 A Simple Interest Rate Rule . . . 41
3.3.2 More Prominent Interest Rate Rules . . . 48
3.4 Conclusion . . . 53
Acknowledgements
I want to thank many people. I thank Jordi Gal´ı for his invaluable advise and for his patience with me. The three chapters of the present thesis are based on my joint work with Tommy Sveen. I thank Tommy for a great time work-ing together. I thank my parents for more thwork-ings than would fit in this page. I also acknowledge that the work presented in this thesis has benefitted from the comments of many of my colleagues. I therefore wish to thank seminar participants at the Annual Congress of the EEA, Central Bank Workshop on Macroeconomic Modelling, IX Workshop on Dynamic Macroeconomics, European University Institute, Norges Bank, and Universitat Pompeu Fabra. Special thanks to Farooq Akram, Mikel Casares, Larry Christiano, Christian Haefke, Jinill Kim, Omar Licandro, Albert Marcet, Martin Menner, Ricardo Reis, Øistein Røisland, Philip Saur´e, Stephanie Schmitt-Groh´e, Jaume Ven-tura, Mike Woodford, and Fredrik Wulfsberg. Finally, I thank Larry Chris-tiano for providing me with some Matlab Code that was useful in some of the computations. Most of the computations are conducted using Dynare http://www.cepremap.cnrs.fr/dynare/. The usual disclaimer applies.
Foreword
How does capital accumulation affect inflation and output dynamics, and what are the consequences for monetary policy? We analyze these ques-tions in a general equilibrium framework with monopolistic competition and staggered price setting.
The first chapter makes progress in explaining the role of capital accu-mulation for inflation and output dynamics. It is assumed that firms face restrictions regarding both price adjustment and capital accumulation. In this sense capital is firm-specific. The main result is that capital accumula-tion affects inflaaccumula-tion dynamics primarily through its impact on the marginal cost. This mechanism is much simpler than the one implied by the analysis of the problem in the earlier related literature. The reason is that the latter has suffered from a conceptual mistake, as we note.
In the second chapter we compare the model with firm-specific capital with an alternative specification where households accumulate capital and rent it to firms. The difference in implied equilibrium dynamics is large, as we justify by proposing a simple metric. This result invites us to interpret some of the puzzling empirical findings that have been obtained using models with staggered price setting and a rental market for capital as an artefact of this particular set of assumptions.
the modeling of capital accumulation. We also identify some desirable prop-erties interest rate rules. Interestingly, the results suggest a reinterpretation of monetary policy under Volcker and Greenspan: the empirically plausible characterization of monetary policy can explain the stabilization of macro-economic outcomes observed in the early eighties for the U.S. economy. The Taylor principle in itself cannot.
In summary, we emphasize the importance of modeling a simultaneous price setting and investment decision at the firm level. Both the positive and the normative conclusions regarding the role of capital in a world with nominal rigidities change if attention is restricted to the price setting decision alone. The latter specification has been the standard modeling choice in the existing literature. This convenience is, however, not innocuous, as we show. Since we wrote and circulated the papers on which my thesis is based other contributions that stress the fruitfulness of assuming firm-specific cap-ital in a model with staggered price setting have mushroomed. I am happy to see the impact of my thesis on the way economists think about capital and prices.
The second and third chapter of the Ph.D. thesis were published in this article: “New Perspectives on Capital, Sticky Prices, and the Taylor Principle”
(with Tommy Sveen), Journal of Economic Theory, 123 (2005), p. 21‐39.
Chapter 1
Pitfalls in the Modeling of
Forward-Looking Price Setting
and Investment Decisions
Abstract:1 The first chapter makes progress in explaining the role of
cap-ital accumulation for inflation and output dynamics. We follow Woodford (2003, Ch. 5) in assuming Calvo pricing, combined with a convex capital adjustment cost at the firm level. The main result is that capital accumula-tion affects inflaaccumula-tion dynamics primarily through its impact on the marginal cost. This mechanism is much simpler than the one implied by the analysis in Woodford’s text. The reason is that his analysis suffers from a concep-tual mistake, as we show. The latter has obscured the economic mechanism through which capital affects inflation and output dynamics in the Calvo model, as discussed in Woodford (2004).
1.1
Introduction
By now there exists a large literature studying business cycles using dynamic New-Keynesian (DNK) models, i.e. stochastic general equilibrium models with imperfect competition and nominal rigidities.2 However, it is generally
assumed that labor is the only productive input, or alternatively, that the
1The first version of the paper on which this chapter is based has been circulated
as Norges Bank Working Paper 2004/1, Oslo, February 11, 2004, http://www.norges-bank.no/publikasjoner/arbeidsnotater/.
capital stock in the economy is constant.3 Woodford (2003, p. 352) comments
on these modeling choices: ‘[...] while this has kept the analysis of the effects of interest rates on aggregate demand quite simple, one may doubt the accuracy of the conclusions obtained, given the obvious importance of variations in investment spending both in business fluctuations generally and in the transmission mechanism for monetary policy in particular.’
DNK models that introduce capital accumulation typically assume a rental market.4 In the present paper we follow Woodford (2003, Chapt. 5) in
assuming firm-specific capital: staggered price setting `a la Calvo is combined with a convex capital adjustment cost at the firm level.5 Along the way we
show that the analysis in Woodford’s text suffers from a conceptual mistake. In a nutshell: he does not assess correctly over what set of future states of the world an optimizing Calvo price setter forms expectations.6
The ultimate goal of the present chapter is to assess the role of endoge-nous firm-specific capital for inflation and output dynamics. To this end we analyze impulse responses to a shock in the exogenous growth rate of money balances7 for two cases: the baseline model with endogenous capital
(hence-forth baseline) and a specification with decreasing returns to scale resulting from a constant capital stock at the firm level (henceforth DRS). We find the
3Erceg et al. (2000) assume a constant aggregate capital stock combined with a rental
market for capital, while Sbordone (2002) and Gal´ı et al. (2001) assume constant capital at the firm level.
4See, e.g., Yun (1996), Smets and Wouters (2003), Christiano et al. (2004), and
Schmitt-Groh´e and Uribe (2004). However, Sveen and Weinke (2003, 2004a,b) show that the rental market assumption is not innocuous in a model with staggered price setting.
5Since we wrote and circulated the first version of the present chapter there have been
other contributions studying firm-specific capital in a Calvo-style model. See, e.g., Altig et al. (2004), Christiano (2004), Eichenbaum and Fisher (2004), and Woodford (2004).
6The same critique applies to Casares (2002).
7In the above mentioned first working paper version of the present chapter we solve
following: first, the response of output is larger in the baseline model – both on impact and during the transition period. Second, the inflation dynamics are similar in the two models.
The intuition is surprisingly simple: first, endogenous capital at the firm level affects inflation dynamics primarily through its impact on the marginal cost. The inflation equation, however, changes only to a negligible extent with respect to the one derived by Sbordone (2002) and Gal´ı et al. (2001) under the assumption that the capital stock is constant at the firm level. Second, there are two opposite effects from capital accumulation on the de-termination of the marginal cost. On the one hand, the additional production triggered by investment demand increases the marginal cost in the baseline model with respect to the DRS specification. On the other hand, the resulting additional capital increases the economy’s productive capacity, thereby de-creasing the marginal cost. The latter is anticipated by forward-looking price setters. This explains why the two models display similar inflation dynam-ics even though the output response is consistently larger with endogenous firm-specific capital.
This mechanism is indeed much simpler than the one outlined in Wood-ford (2003, Ch. 5). His analysis implies that firm-specific capital combined with Calvo pricing results in a substantial change in the dynamic relation-ship between the average real marginal cost and inflation. This has obscured the economic mechanism through which capital affects inflation and output dynamics, as discussed in Woodford (2004).8
The remainder of the chapter is organized as follows: Section 1.2 outlines the baseline model. In particular, it is shown why the price setting
prob-8One particularly problematic feature of the inflation equation in Woodford (2003, Ch.
lem associated with that structure has not been solved in a correct way in Woodford (2003, Ch. 5). In Section 1.3 we present and interpret our results. Section 1.4 concludes.
1.2
The Model
We follow the general equilibrium structure outlined in Woodford (2003, Ch. 5).9 There are two sectors, households and firms. Households choose labor
supply and consumption demand. They have access to complete financial markets and supply labor in a perfectly competitive market. Firms produce differentiated goods and act under monopolistic competition. They face re-strictions on both price adjustment and capital accumulation.
The only aggregate uncertainty comes from the growth rate of money balances, which we assume to follow an AR(1) process:
∆mt=ρm∆mt−1+εt, (1.1)
where ∆ denotes the difference operator, and mt is the log of nominal money
balances Mt at time t. The autoregressive parameter, ρm, is assumed to be
strictly positive and less than one. Finally, εt is assumed to beiid with zero
mean and variance σ2
ε.
1.2.1
Households
A representative household maximizes expected discounted utility:
Et ∞
X
k=0
βkU(C
t+k, Nt+k), (1.2)
9His model is more general than ours. However, this is irrelevant for our dicussion of
whereEtdenotes the expectational operator conditional on information
avail-able up to time t. Furthermore, U(·) is period utility, and parameter β is the discount factor. Hours worked in period t are denoted Nt. Finally,
Ct ≡
³R1
0 Ct(i)
ε−1
ε di
´ ε
ε−1
denotes the time t Dixit-Stiglitz consumption ag-gregator, and parameter ε is the elasticity of substitution between different varieties of goods Ct(i).
The maximization is subject to a sequence of budget constraints:
Z 1
0
Pt(i)Ct(i)di+Et{Qt,t+1Dt+1} ≤Dt+WtNt+Tt, (1.3)
where Qt,t+1 and Dt+1 denote, respectively, the stochastic discount factor
for random nominal payments and the nominal payoff associated with the portfolio held at the end of periodt. Moreover, Pt(i) gives the nominal price
of varietyiat timet,Wtis the nominal wage as of that period, andTtdenotes
profits resulting from ownership of firms. We assume a standard period utility:
U(Ct, Nt) =
Ct1−σ 1−σ −
Nt1+φ
1 +φ, (1.4)
where parameterσ denotes household’s relative risk aversion, and parameter φ can be interpreted as the the inverse of the aggregate Frisch labor supply elasticity.
Cost minimization by households implies that for each variety of goods the consumption demand function reads:
Cd t (i) =
µ Pt(i)
Pt
¶−ε
Ct, (1.5)
where Pt ≡
³R1
0 Pt(i)
1−εdi´
1 1−ε
The remaining first order conditions associated with the household’s prob-lem are as follows:
Cσ
tNtφ =
Wt
Pt
, (1.6)
β µ
Ct+1
Ct
¶−σµ Pt
Pt+1
¶
= Qt,t+1. (1.7)
The first equation is the optimality condition for labor supply, and the second is a standard intertemporal optimality condition. Finally, let us note that the price of a risk-less one-period bond is given by R−1
t =EtQt,t+1, whereRt
denotes the gross nominal interest rate.
1.2.2
Firms
Firms are indexed on the unit interval. Each firm has access to a Cobb-Douglas production technology:
Yt(i) = Kt(i)αNt(i)1−α, (1.8)
whereKt(i) and Nt(i) denote, respectively, capital holdings and labor input
used by firm i in its period t production denoted Yt(i). Parameter α is the
capital share.
Each firm i makes an investment decision at any point in time with the resulting additional capital becoming productive one period after the in-vestment decision is made. As in Woodford (2003, Ch. 5) we assume the following: first, the investment good is a Dixit-Stiglitz aggregate of all of the goods in the economy with the same constant elasticity of substitution as in the consumption aggregate. Second, firms face a convex adjustment cost of changing their capital holdings. Given firm i’s time t capital stockKt(i) the
at this point in time in order to have a capital stock Kt+1(i) in place in the
next period is given by:
It(i) = I
µ
Kt+1(i)
Kt(i)
¶
Kt(i). (1.9)
Function I(·) has the following characteristics: I(1) = δ, I0(1) = 1, and
I00(1) = ²
ψ. Parameter δ denotes the depreciation rate. Eichenbaum and
Fisher (2004) interpret parameter ²ψ as the elasticity of the investment to
capital ratio with respect to Tobin’s q, evaluated in steady state. Parame-ter ²ψ is assumed to be strictly larger than zero and it measures the convex
capital adjustment cost in a log-linear approximation to the equilibrium dy-namics.
Firms post sticky prices `a la Calvo (1983), i.e. each period a measure (1−θ) is randomly selected. Those firms change their prices and the re-maining firms post their last period’s nominal prices. Cost minimization by firms and households implies that demand for each individual good i in period t can be written as follows:
Ytd(i) = µ
Pt(i)
Pt
¶−ε
Ytd, (1.10)
where Yd
t ≡Ct+It denotes aggregate time t demand, and It≡
R1
0 It(i)di is
aggregate time t investment demand.
With probabilityθk a price that was chosen at time t will still be posted
at timet+k. When setting a new priceP∗
t(i) in periodtfirmimaximizes the
for©P∗
t+k(i), Kt+k+1(i), Nt+k(i)
ª∞
k=0in order to solve the following problem: 10 max ∞ X k=0 Et © Qt,t+k
£ Yd
t+k(i)Pt+k(i)−Wt+kNt+k(i)−Pt+kIt+k(i)
¤ª
(1.11)
s.t.
Yd
t+k(i) =
µ
Pt+k(i)
Pt+k
¶−ε Yd
t+k,
Yd
t+k(i) ≤ Nt+k(i)1−αKt+k(i)α,
It+k(i) = I
µ
Kt+k+1(i)
Kt+k(i)
¶
Kt+k(i),
Pt+k+1(i) =
½ P∗
t+k+1(i) with prob. (1−θ)
Pt+k(i) with prob. θ
The implied first order condition for capital accumulation reads:
dIt(i)
dKt+1(i)
Pt=Et
½ Qt,t+1
·
MSt+1(i)−
dIt+1(i)
dKt+1(i)
Pt+1
¸¾
, (1.12)
where MSt+1(i) denotes the nominal marginal savings in firm i’s labor cost
associated with having one additional unit of capital in place in period t+ 1. The intuition behind the last equation is the following: the marginal cost of installing an additional unit of capital at time t (including the adjustment cost) is equalized to the expected discounted marginal contribution to the firm’s value associated with having that additional unit of capital in place at point in timet+1. The latter is given by the marginal return from using it for production, MSt+1(i), and selling the remaining capital after depreciation
(net of the change in the time t+ 1 adjustment cost that is associated with the timetinvestment decision). As has been emphasized by Woodford (2003, Ch. 5), the relevant measure of the marginal return to capital is the marginal savings in a firm’s labor cost: firms are demand constrained and hence the
10A firmjthat is restricted to change its price at timetsolves the same problem, except
return from having an additional unit of capital in place results from the fact that this allows to produce the quantity that happens to be demanded using less labor.
The following relationship holds true:
MSt(i) =Wt
MP Kt(i)
MP Lt(i)
, (1.13)
whereMP Kt(i) and MP Lt(i) denote, respectively, the marginal product of
capital and labor of firm i in period t.
The first order condition for price setting is given by:
∞
X
k=0
θkE t
©
Qt,t+kYtd+k(i) [Pt∗(i)−µMCt+k(i)]
ª
= 0, (1.14)
where µ≡ ε
ε−1 is the frictionless mark-up over marginal costs, and MCt(i)
denotes the nominal marginal cost of firm i in period t. The latter is given by:
MCt(i) =
Wt
MP Lt(i)
. (1.15)
Equation (1.14) is the familiar first order condition implied by the Calvo model: optimizing price setters behave in a forward-looking manner, i.e. they take into account not only current but also future expected marginal costs in those states of the world where the chosen price is still posted.11
The only non-standard feature in equation (1.14) is that capital affects labor productivity and hence a firm’s marginal cost. This aspect of a firm’s price setting decision results in an intricate problem. As we argue next, the latter has not been solved in a correct way in Woodford (2003, Ch. 5).
11We follow a large literature on the Calvo model in using the notationE
tin equation (1.14) to indicate an expectation that is conditional on the timet state of the world, but integrating only over those future states in which firmihas not reset its price since period
t. Woodford (2004) usesEbi
1.2.3
A Short Note on Woodford’s Conceptual
Mis-take
To fix ideas we represent firm i’s price setting problem at timet by a simple tree, which consists of the states of the world that are consistent with the current state S. This is shown in Figure 1. Equations (1.14) and (1.15) prescribe that the relevant capital holdings are associated with those states of the world where the newly set price is still posted. We refer to these states as the Calvo states. In Figure 1 they are assumed to correspond to nodesS, S0, S00,... in the tree. Firm i’s capital stock at nodeS is predetermined.
Time t
Time t+1
Time t+2
S
S0 S1
S00 S01 S10 S11
θ
θ θ
1−θ
1−θ 1−θ
.
.
.
.
.
.
.
Figure 1: Price setting with firm-specific capital accumulation.
time t expectation regarding its futureoptimally chosen prices. Specifically, he restricts attention to firmi’s timetexpectation of its future relative prices in the Calvo states. This is not correct, as we show next.
Clearly, it is enough to show that firmi’s timetexpectation regarding one of its future capital holdings in the Calvo states is computed in an incorrect way. To this end we consider firmi’s timetchoice of its next period’s capital stock. Equations (1.12) and (1.13) state that this choice takes rationally into account that firm i’s time t+ 1 price might be optimally chosen. But this means that the possibility of choosing a new price in period t+ 1 affects a price setter’s timetinvestment decision and hence its timet+1 capital stock,
in particular, if node S0 is reached at point in time t+ 1. Therefore, firm i’s time t expectation regarding its capital holdings in the Calvo states does depend on its time texpectation regarding future optimally chosen prices, as we have claimed.
1.2.4
Market Clearing
Clearing of the labor market requires that hours worked, Nt, are given by
the following equation, which holds for all t:
Nt=
Z 1
0
Nt(i)di. (1.16)
Moreover, it is useful to define time taggregate capitalKt ≡
R1
0 Kt(i)di and
auxiliary variable Yt ≡KtαNt1−α.12
For each variety isupply, Yt(i), must equal demand:
Yt(i) =Ctd(i) +Itd(i), (1.17)
where Id
t (i) denotes investment demand for good i.
12The difference betweenY
1.2.5
Linearized Equilibrium Conditions
We restrict attention to a log-linear approximation to the equilibrium dynam-ics around a steady state with zero inflation. In what follows, the percent deviation of a variable with respect to its steady state value is denoted by a hat.
Households
Log-linearizing and rearranging the first order condition (1.7) we obtain the household’s Euler equation:
b
Ct=EtCbt+1−
1
σ(it−Etπt+1−ρ), (1.18)
where it denotes the timet nominal interest rate, and πt ≡log
³
Pt
Pt−1
´ is the rate of inflation. Finally, the time discount rate is given by ρ≡ −logβ.
Log-linearizing the household’s labor supply equation (1.6) results in:
d µ
Wt
Pt
¶
=φNbt+σCbt. (1.19)
For convenience, we just assume a standard demand for real balances Mt
Pt:
d µ
Mt
Pt
¶
=Ybt−η(it−ρ), (1.20)
where parameter η denotes the semi-elastisity of demand for real balances with respect to the nominal interest rate.
Firms
We log-linearize the first order condition for investment (1.12) and average over all firms in the economy.13 Combining the resulting relationship with the
Euler equation (1.18) we obtain the following law of motion of the aggregate capital stock:
∆Kt+1 = βEt∆Kt+2+
1 ²ψ
Et{(1−β(1−δ))msct+1
−(it−Etπt+1−ρ)}. (1.21)
where mst ≡
R1
0
M St(i)
Pt di denotes the average real marginal savings in labor costs at time t.
The inflation equation is derived from averaging optimal price setting decisions and aggregating prices via the price index. A natural starting point is the log-linearized real marginal cost at the firm level. The latter reads:
c
mct(i) =mcct−
εα
1−αpbt(i)− α
1−αbkt(i), (1.22) where kt(i) ≡ KKt(ti) and mct ≡
R1
0
M Ct(i)
Pt di denotes the average time t real marginal cost in the economy.
We refer to bkt(i) as firm i’s capital gap at time t. The intuition behind
equation (1.22) is the following: for a zero capital gap a firm that posts a higher than average price faces a lower than average marginal cost due to the implied increase in its marginal product of labor. This is reflected in the second term, and it is exactly as in Sbordone (2002) and Gal´ı et al. (2001) for models with decreasing returns to scale and labor as the only variable input in production. With capital accumulation there is an extra effect coming from the firm’s capital stock, which corresponds to the last term. Conditional on posting the average price in the economy a firm that has a higher than average capital stock in place faces a lower than average marginal cost. The reason is that the marginal product of labor increases with the capital stock used by the firm.
at time t, p∗
t (i)≡ P ∗
t(i)
Pt , can be log-linearized as:
b p∗
t(i) = ∞
X
k=1
(βθ)kEtπt+k+ξ ∞
X
k=0
(βθ)kEtmcct+k−ψ ∞
X
k=0
(βθ)kEtbkt+k(i),
(1.23) where ξ ≡ (1−1−βθα)(1+εα−α), and ψ ≡ (11−−αβθ+εα)α.14 Hence, in addition to the usual
inflation and average marginal cost terms a firm’s optimal price setting de-cision does also depend on its current and future expected capital gaps over the (random) lifetime of the chosen price.
Woodford (2004) shows that the associated inflation equation takes the following simple form:
πt =βEtπt+1+κ mcct, (1.24)
where κ is a parameter which he computes numerically.15
Finally, we note that the aggregate production function is given by:
b
Yt =αKbt+ (1−α)Nbt. (1.25)
Market clearing
Since equation (1.17) holds for each variety in the economy we are entitled to integrate on both sides. Log-linearizing the resulting equation we obtain the aggregate goods market clearing condition:
b
Yt=ζCbt+ (1−ζ)1
δ h
b
Kt+1−(1−δ)Kbt
i
, (1.26)
where ζ ≡ ρ+δρ(1+δ−α) denotes the steady state consumption to output ratio. The steady state capital to output ratio is given by (1−ζ)1
δ.
14The variables in (1.23) are constant in the steady state.
1.3
Equilibrium Dynamics
Given the specification of monetary policy in (1.1), the equilibrium processes for the nominal interest rate, consumption, real wage, real balances, capital, inflation, output, and hours are given by equations (1.18), (1.19), (1.20), (1.21), (1.24), (1.25), and (1.26). The relevant average marginal products entering the average real marginal cost and marginal savings in labor costs are obtained fromYt ≡KtαNt1−α. We analyze impulse responses to a positive
one standard deviation shock in the growth rate of money balances.
1.3.1
Calibration
The period length is one quarter. We choose ²ψ = 3, as suggested by
Wood-ford (2003, Ch. 5) and the references herein. The intertemporal elasticity of substitution is given by 1
σ. Assuming σ = 2 is in line with empirical
es-timates.16 Consistent with a unit labor supply elasticity, we assume φ = 1.
The semi-elasticity of demand for real balances with respect to the nominal interest rate, η, is set to unity implying an empirically plausible value of about 0.05 for the interest rate elasticity. The capital share in the produc-tion funcproduc-tion, α, is 0.36. We set β = 0.99 implying an average annual real return of about 4 percent. Setting θ = 0.75 means that the average lifetime of a price is equal to one year. Consistent with the estimated autoregressive process for M1 in the United States we assume ρm = 0.5 and σ2ε = 0.1.17
Setting ε= 11 implies a frictionless markup of 10 percent.18
16See, e.g., Basu and Kimball (2003) and the references herein. 17Our calibration of parameters φ, α, β, θ, ρ
m, and σ2ε is justified in Gal´ı (2003) and the references herein.
1.3.2
Results
We compare the responses to a monetary policy shock for the baseline model and a specification with decreasing returns to scale resulting from a constant capital stock at the firm level. The result is shown is Figure 2: first, output is higher in the former – both on impact and during the transition period. Second, the inflation dynamics are similar in the two models.
0 5 10 15 20 25 30 35 40
0 0.05 0.1 0.15 0.2 0.25
Output
Baseline DRS
0 5 10 15 20 25 30 35 40
0 0.01 0.02 0.03 0.04
Inflation
Baseline DRS
Let us develop the intuition behind our result. We start by observing that firm-specific capital affects inflation dynamics primarily through its impact on the marginal cost. The form of the inflation equation, however, is only affected to some negligible extent by the feature of capital accumulation at the firm level: if κ in equation (1.24) is approximated by the coefficient premultiplying the marginal cost in the inflation equation associated with the DRS specification,19 then the resulting loss in accuracy is negligible, as
shown in Figure 3.20 The reason is as follows. To the extent that there exists
a capital adjustment cost the firm’s investment decision is forward-looking. If the planning horizon for the investment decision is long enough, then price setters and non-price setters do not make very different investment decisions, on average. The fact that they face the same probabilities of being allowed or restricted to change their prices over the relevant planning horizon leads to a small difference in their current investment decisions and, more generally, in their expected investment policies.
Next we note that there are two counteracting effects from capital ac-cumulation on the determination of the marginal cost. On the one hand, investment spending adds to aggregate demand, thereby implying higher production and an increase in the marginal cost in response to the shock. On the other hand, the additional capital resulting from investment spend-ing in one period increases the economy’s productive capacity in subsequent periods. This implies a decrease in marginal costs.
19Sbordone (2002) and Gal´ı et al. (2001) show that this coefficient takes the following
form: (1−βθθ)(1−θ) 1−α
1+α(ε−1).
20We acknowledge a tiny difference between the baseline impulse responses reported in
0 10 20 30 40 0
0.01 0.02 0.03
0.04 Inflation
Baseline Approx
0 10 20 30 40
0 0.05 0.1 0.15 0.2
0.25 Output
Baseline Approx
0 10 20 30 40
−5 0 5 10 15
20x 10
−4 Inflation
Difference
0 10 20 30 40
−6 −4 −2
0x 10
−3 Output
Difference
Figure 3: Capital works through the marginal cost.
1.4
Conclusion
The present chapter makes progress in explaining the economic mechanism through which capital accumulation affects inflation and output dynamics. We use a Calvo-style model with a convex capital adjustment cost at the firm level. Our main finding is that firm-specific capital accumulation affects primarily the determination of the marginal cost. The form of the infla-tion equainfla-tion, however, changes only to a negligible extent compared with a model where the capital stock at the firm level is assumed to be constant. Combined with the fact that investment demand has counteracting effects on the determination of the marginal cost this leads to a surprisingly simple intuition for the associated inflation and output dynamics. This economic mechanism has been obscured by a conceptual mistake in Woodford (2003, Ch. 5), as we show.
Appendix: Inflation Dynamics
Woodford (2004) posits that the price chosen by a Calvo price setter i is:
b p∗
t(i) =pb∗t −τ1bkt(i), (A1)
where τ1 is an unknown parameter. He further assumes that the investment
decision of any firm j satisfies:
b
kt+1(j) = τ2bkt(j) +τ3pbt(j), (A2)
where τ2 and τ3 are two additional unknown parameters.
Finally, he invokes the relationship between the log-linearized average newly set price, pb∗
t, and inflation, πt:
πt =
1−θ θ pb
∗
t. (A3)
Combined with the first-order conditions for price setting and investment it is possible to pin down the unknown coefficients τ1, τ2, and τ3 and to derive
References
Altig, David, Lawrence J. Christiano, Martin Eichenbaum, and Jesper Linde (2004): “Firm-Specific Capital, Nominal Rigidities, and the Business Cycle”, mimeo.
Basu, Susanto, and Miles S. Kimball (2003): “Investment Planning Costs and the Effects of Fiscal and Monetary Policy”, mimeo.
Calvo, Guillermo (1983): “Staggered Prices in a Utility Maximizing Frame-work”, Journal of Monetary Economics, 12(3), 383-398.
Casares, Miguel (2002), “Time-to-Build Approach in a Sticky Price, Sticky Wage Optimizing Monetary Model”, European Central Bank Working Paper No. 147.
Chari, V. V., Patrick J. Kehoe, and Ellen R. McGratten (2000): “Sticky Price Models of the Business Cycle: Can the Contract Multiplier Solve the Persistence Problem”, Econometrica, 68, 1151-1179.
Christiano, Lawrence J. (2004): “Firm-Specific Capital and Aggregate Infla-tion Dynamics in Woodford’s Model”, mimeo.
Christiano, Lawrence J., Martin Eichenbaum, and Charles Evans (2004): “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy”, forthcoming in Journal of Political Economy.
Clarida, Richard, Jordi Gal´ı, and Mark Gertler (1999): “The Science of Mon-etary Policy: a New Keynesian Perspective”,Journal of Economic Literature, 37(4), 1661-1707.
Erceg, Christopher J., Dale W. Henderson, and Andrew T. Levin (2000): “Optimal Monetary Policy with Staggered Wage and Price Contracts”, Jour-nal of Monetary Economics, 46(2), 281-313.
Gal´ı, Jordi (2003): “New Perspectives on Monetary Policy, Inflation, and the Business Cycle”, in M. Dewatripont, L. Hansen, and S. Turnovsky (editors),
Advances in Economics and Econometrics, Volume III, 151-197, Cambridge University Press.
Gal´ı, Jordi, Mark Gertler, and David L´opez-Salido (2001): “European Infla-tion Dynamics”, European Economic Review, 45(7), 1237-1270.
Sbordone, Argia M. (2002): “Prices and Unit Labor Costs: A New Test of Price Stickiness”, Journal of Monetary Economics, 49, 265-292.
Smets, F. and R. Wouters (2003): “An Estimated Stochastic Dynamic Gen-eral Equilibrium Model of the Euro Area”,Journal of the European Economic Association, 1, 1123-1175.
Schmitt-Groh´e, Stephanie, and Martin Uribe (2004): “Optimal Simple and Implementable Monetary and Fiscal Rules”, NBER Working Paper 10253.
Sveen, Tommy, and Lutz Weinke (2003): “Inflation and Output Dynamics with Firm-owned Capital”, Universitat Pompeu Fabra Working Paper No. 702.
Sveen, Tommy, and Lutz Weinke (2004a): “New Perspectives on Capital and Sticky Prices”, Norges Bank Working Paper 2004/3.
Sveen, Tommy, and Lutz Weinke (2004b): “Firm-Specific Investment, Sticky Prices, and the Taylor Principle”, Norges Bank Working Paper 2004/12.
Chapter 2
New Perspectives on Capital
and Sticky Prices
Abstract:1 We model capital accumulation in a dynamic New-Keynesian
model with staggered price setting `a la Calvo. Capital is assumed to be firm-specific, as analyzed in the first chapter. We now compare this model with an alternative specification where households accumulate capital and rent it to firms. The difference in implied equilibrium dynamics is large, as justified by a simple metric. This result invites us to interpret some of the puzzling empirical findings that have been obtained using models with staggered price settingand a rental market for capital as an artefact of this particular set of assumptions.
2.1
Introduction
In the field of New-Keynesian macroeconomics there has been recent interest in models with staggered price setting that allow for capital accumulation.2
The main reason is that many research questions can only be addressed if capital accumulation is taken into account.3 Moreover, it has been argued
1The first version of the paper on which this chapter is based has been circulated
as Norges Bank Working Paper 2004/3, Oslo, February 23, 2004, http://www.norges-bank.no/publikasjoner/arbeidsnotater/.
2For an early contribution see, e.g., Yun (1996).
3See, e.g., Gal´ı et al. (2004). The authors consider rule-of-thumb consumers in addition
that modeling investment demand might help explain some empirical regu-larities once additional features are introduced into the model, which would be hard to entertain if consumption was the only component of aggregate demand.4 However, it is unclear a priori how capital accumulation should
be introduced into such a model. As has been argued by Woodford (2003, Ch. 5), combining the assumptions of staggered price setting and a rental market for capital is convenient but potentially unappealing: it affects the determination of the marginal cost at the firm level in a non-trivial way. Our understanding of New-Keynesian models with staggered price setting and capital accumulation is therefore obscured as long as the quantitative consequences of the widely used rental market assumption remain opaque.
The present chapter fills that gap in the existing literature: the rental market case is compared with a baseline model with firm-specific capital. In both models staggered price setting `a la Calvo is combined with the following (standard) restrictions on capital formation: the additional capital resulting from an investment decision becomes productive with a one period delay, and there is a convex adjustment cost in the process of capital accumulation. The two models are compared in a simulation exercise where we analyze the respective impulse responses to a shock in the exogenous growth rate of money balances.
The main finding is the following: for any given restriction on price ad-justment there is a substantial amount of additional price stickiness in the baseline model compared with the rental market specification. We justify this claim by proposing a metric, which gives a precise quantitative meaning
4Christiano et al. (2004) and Smets and Wouters (2003) use the assumption of
to it. The intuition behind our result is plain from a comparison of the price setters in the two models: with a restriction on capital adjustment at the firm level, as in the baseline model, an increase in a firm’s price is associated with a decrease in its marginal cost.5 We refer to this feature of the baseline
model as short run decreasing returns to scale. This effect is absent if a rental market for capital is assumed. The latter implies that each firm in the economy faces the same marginal cost, which is independent of the quan-tity supplied by any individual firm. This mechanism has been discussed by Sbordone (2002) and Gal´ı et al. (2001) for models with decreasing returns to scale resulting from a fixed capital stock at the firm level.6 Our work shows
that short run decreasing returns to scale in the baseline model suffice to imply equilibrium dynamics that are quantitatively different from the ones associated with the rental market specification. The different price setting incentives in the two models are indeed the driving force behind our result: the only difference between the two models lies in the characterization of the respective inflation dynamics.7
It is obvious that the theoretical insights developed in this chapter can be used to explain why the econometrician tends to overestimate the price stick-iness in actual economies if aggregate data are analyzed through the lense of a dynamic New-Keynesian model featuring a rental market for capital. Therefore not surprisingly, Altig et al. (2004) and Eichenbaum and Fisher (2004) find that assuming firm-specific capital results in estimated values of
5In the baseline model we assume that the capital stock at the firm level is
prede-termined and that there exists a capital adjustment cost. One of the two assumptions
would suffice to imply that a firm’s price setting decision affects its marginal cost. The role of a predetermined capital stock at the firm levelper se, i.e. abstracting from capital adjustment costs, has been analyzed by Sveen and Weinke (2003).
6See Woodford (1996) for an early model with differences in marginal costs among
producers.
7The latter holds up to the first order approximation to the equilibrium dynamics,
the Calvo parameter that appear to be in line with the micro evidence. It also follows from our results that some of the puzzling empirical findings in Christiano et al. (2004) and Smets and Wouters (2003) might be interpreted as an artefact of assuming staggered price setting combined with a rental market for capital.
The remainder of the chapter is organized as follows: Section 2.2 outlines the baseline model and the rental market specification. In Section 2.3 we conduct the above mentioned simulation exercise and interpret the results. Section 2.4 concludes.
2.2
Theoretical Framework
2.2.1
Firm-Specific Capital
The model with firm-specific capital has already been developed in the first chapter. As a reminder we just summarize the relevant log-linearized equi-librium conditions. The household’s Euler equation:
b
Ct=EtCbt+1−
1
σ(it−Etπt+1−ρ). (2.1)
The household’s labor supply equation:
d µ
Wt
Pt
¶
=φNbt+σCbt. (2.2)
Demand for real balances: d µ
Mt
Pt
¶
=Ybt−η(it−ρ). (2.3)
Law of motion of the aggregate capital stock:
∆Kt+1 = βEt∆Kt+2+
1 ²ψ
Et{(1−β(1−δ))msct+1
Inflation equation:
πt =βEtπt+1+κ mcct. (2.5)
Aggregate production function:
b
Yt =αKbt+ (1−α)Nbt. (2.6)
Aggregate goods market clearing:
b
Yt=ζCbt+ (1−ζ)
1 δ
h b
Kt+1−(1−δ)Kbt
i
. (2.7)
Monetary Policy:
∆mt=ρm∆mt−1+εt. (2.8)
Again, the marginal products entering the average real marginal cost and marginal savings in labor costs are to be obtained from Yt≡KtαNt1−α.
2.2.2
Rental Market for Capital
We now assume that the representative household accumulates the capital stock and rents it to intermediate goods firms. This is the only change with respect to the specification with firm-specific capital. The household maxi-mizes the objective function given in (1.2) subject to the following sequences of constraints:
Pt(Ct+It) +Et{Qt,t+1Dt+1} ≤ Dt+WtNt+RktKt+Tt, (2.9)
It = I
µ Kt+1
Kt
¶
Kt. (2.10)
Again, function I(·) is assumed to have the characteristics outlined in the first chapter. Rk
t denotes the time t rental rate of capital. Hence, RktKt is
The first order conditions associated with the household’s choices over leisure and the time path of consumption are identical to the ones given in equations (1.6) and (1.7), respectively. The first order condition associated with the household’s investment decision is:
dIt
dKt+1
Pt =Et
½ Qt,t+1
·
Rkt+1− dIt+1
dKt+1
Pt+1
¸¾
. (2.11)
Cost minimization implies that each firm produces at the same capital labor ratio. The marginal cost is therefore common to all firms, and this allows us to write the rental rate of capital as follows:
Rk t =Wt
MP Kt
MP Lt
. (2.12)
Log-linearizing equation (2.11) and invoking (2.1) we recover the same log-linearized law of motion of capital as the one given in equation (2.4). This means that, up to a log-linear approximation to the equilibrium dynamics, the set of equilibrium conditions is identical to the one associated with the baseline model, except for the inflation equation: with a rental market for capital a firm’s marginal cost is independent of its price setting decision. The resulting inflation equation therefore takes the following standard form:
πt =βEtπt+1+λmcct, (2.13)
where λ ≡ (1−βθθ)(1−θ), and the average marginal cost is defined in the same way as in the baseline model.8
8See Gal´ı (2004) et al. for a detailed development of a Calvo- style model with a rental
2.3
Results
Both the baseline model with firm-specific capital and the alternative rental market specification are calibrated as discussed in the first chapter.9
For both models we analyze impulse responses associated with a positive one standard deviation shock to the growth rate of money balances.
0 5 10 15 20 25
0 0.02 0.04 0.06 0.08 0.1
Inflation
0 5 10 15 20 25
0 0.05 0.1 0.15 0.2 0.25
Output
Baseline Rental Market Baseline Rental Market
Figure 1: Endogenous firm-specific capital vs. rental market.
9It should be noted that in both cases we need to assign values to exactly the same set
The inflation response to the shock is relatively smaller on impact in the baseline model. However, it becomes eventually larger than the correspond-ing level in the rental market specification. Moreover, the output reaction is larger in the baseline model both on impact and during the transition. This is shown in Figure 1. The intuition is as follows: to the extent that prices are sticky a positive monetary policy shock affects real interest rates and stimulates aggregate demand. This implies an increase in current and future expected marginal costs. Without a rental market for capital a price setter is more reluctant to change its price in response to the shock. The reason is that the firm takes into account that its marginal cost is affected, to some extent, by the chosen price: due to the restrictions on a firm’s capital adjustment a price increase is associated with a decrease in its marginal cost. This effect is absent if a rental market for capital is assumed. In that case each firm produces at the same marginal cost, which is independent of the quantity an individual firm supplies. This means that for any given restric-tion on price adjustment there is addirestric-tional price stickiness in the baseline model with respect to the rental market specification.
model for empirical analysis would tend to overestimate the degree of price stickiness. For instance, Smets and Wouters (2003) amend their empirical analysis with a caveat of this kind. Their estimate of the expected lifetime of a price is two and a half years, which is far fetched. Our theoretical re-sult shows that this somewhat puzzling finding might reflect the quantitative consequences of the rental market assumption. Our result sheds also light on a finding by Christiano et al. (2004). Their empirical estimate of the price stickiness parameter in a Calvo-style model with capital accumulation and a rental market is ‘driven to unity’. They claim that this is an unappealing feature of sticky price models. However, we tend to interpret their finding as an artefact of the rental market assumption.10 From our theoretical results it
is also plain that the estimated price stickiness must be considerably smaller, if the aggregate data are analyzed using a DNK model with firm-specific capital. This is confirmed by the empirical results in Altig et al. (2004) and Eichenbaum and Fisher (2004).11
Of course, the adjustment of the price stickiness parameter that is needed in the rental market model in order to generate the same equilibrium dynam-ics as in the baseline model depends on the calibration. This is shown in Fig-ure 2. First, if the elasticity of substitution between goods, ε, increases then a price setter is more reluctant to change its price in the baseline model. The reason is that a higher value of ε implies that a firm’s price setting decision has a stronger impact on its marginal cost. Therefore, more price stickiness
10It should be noticed, however, that both Smets and Wouters (2003) and Christiano et
al. (2004) assume an investment adjustment cost combined with other features that are not present in the models we compare in the present paper.
11It should be noted, however, that both Altig et al. (2004) and Eichenbaum and Fisher
is needed in the rental market model in order to make the two impulse re-sponses coincide. This is shown in the upper panel of Figure 2. Second, an increase in the capital share in the production function, α, has a similar effect: it increases the price setters’ reluctance to change their prices in the baseline model. As is shown in the lower panel of Figure 2, the latter implies that more price stickiness is needed in the rental market model in order to generate the same equilibrium dynamics as in the baseline model.
0 2 4 6 8 10 12 14 16 18 20
0.75 0.8 0.85 0.9 0.95 1
ε
Metric
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.75
0.8 0.85 0.9 0.95 1
α
Metric
2.4
Conclusion
We should emphasize the main contribution of the present chapter and some of the issues that are left for future research. We analyze New-Keynesian models with staggered price setting `a la Calvo and a convex adjustment cost in the process of capital accumulation. In the baseline model it is assumed that firms do not have access to a rental market for capital. We compare this model with an alternative specification where a rental market is assumed. Our main finding is that the difference in implied equilibrium dynamics is large and we propose a metric, which gives a precise quantitative meaning to that statement. This theoretical result sheds light on some of the puzzling empirical findings that have been obtained using New-Keynesian models with staggered price setting and a rental market for capital.
References
Altig, David, Lawrence J. Christiano, Martin Eichenbaum, and Jesper Lind´e (2004): “Firm-Specific Capital, Nominal Rigidities, and the Business Cycle”, mimeo.
Basu, Susanto and Miles S. Kimball (2003): “Investment Planning Costs and the Effects of Fiscal and Monetary Policy”, mimeo.
Calvo, Guillermo (1983): “Staggered Prices in a Utility Maximizing Frame-work”, Journal of Monetary Economics, 12(3), 383-398.
Casares, Miguel (2002): “Time-to-Build Approach in a Sticky Price, Sticky Wage Optimizing Monetary Model”, European Central Bank Working Paper No. 147.
Christiano, Lawrence J., Martin Eichenbaum, and Charles Evans (2004): “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy”, forthcoming in Journal of Political Economy.
Clarida, Richard, Jordi Gal´ı, and Mark Gertler (1999): “The Science of Monetary Policy: A New Keynesian Perspective”, Journal of Economic Lit-erature, 37(4), 1661-1707.
Edge, Rochelle M. (2000): “Time-to-Build, Time-to-Plan, Habit Persistence, and the Liqidity Effect”, International Finance Discussion Papers No. 673.
Eichenbaum, Martin and Jonas D. M. Fisher (2004): “Evaluating the Calvo Model of Sticky Prices”, NBER Working Paper 10617.
Gal´ı, Jordi (2003): “New Perspectives on Monetary Policy, Inflation, and the Business Cycle”, in M. Dewatripont, L. Hansen, and S. Turnovsky (editors),
Advances in Economics and Econometrics, Volume III, 151-197, Cambridge University Press.
Gal´ı, Jordi, Mark Gertler, and David L´opez-Salido (2001): “European Infla-tion Dynamics”, European Economic Review, (45)7, 1237-1270.
Gal´ı, Jordi, David L´opez-Salido, and Javier Vall´es (2004): “Rule-of-Thumb Consumers and the Design of Interest Rate Rules”, forthcoming in Journal of Money, Credit, and Banking.
Sbordone, Argia M. (2002): “Prices and Unit Labor Costs: A New Test of Price Stickiness”, Journal of Monetary Economics, 49, 265-292.
Smets, Frank and Raf Wouters (2003): “An Estimated Stochastic Dynamic General Equilibrium Model of the Euro Area”,Journal of the European Eco-nomic Association, 1(5), 1123-1175.
Sveen, Tommy and Lutz Weinke (2003): “Inflation and Output Dynamics with Firm-owned Capital”, Universitat Pompeu Fabra Working Paper No. 702.
Sveen, Tommy and Lutz Weinke (2004): “Pitfalls in the Modeling of Forward-Looking Price Setting and Investment Decisions”, Norges Bank Working Pa-per No. 2004/1.
Woodford, Michael (1996): “Control of Public Debt: A Requirement for Price Stability?”, NBER Working Paper No. 5684.
Woodford, Michael (2003): Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press.
Chapter 3
Firm-Specific Investment,
Sticky Prices and the Taylor
Principle
Abstract:1 According to the Taylor principle a central bank should adjust
the nominal interest rate by more than one-for-one in response to changes in current inflation. Most of the existing literature supports the view that by following this simple recommendation a central bank can avoid being a source of unnecessary fluctuations in economic activity. The present chapter shows that this conclusion is not robust with respect to the modeling of capital accumulation. We use our insights to discuss the desirability of alternative interest rate rules. Our results suggest a reinterpretation of monetary policy under Volcker and Greenspan: The empirically plausible characterization of monetary policy can explain the stabilization of macroeconomic outcomes observed in the early eighties for the US economy. The Taylor principle in itself cannot.
3.1
Introduction
According to the Taylor principle a central bank should follow an active monetary policy, i.e. it should adjust the nominal interest rate by more than one-for-one in response to changes in current inflation. Simple interest rate
1The first version of the paper on which this chapter is based has been circulated
rules consistent with that recommendation guarantee determinacy, i.e. local uniqueness of rational expectations equilibrium (REE), in many dynamic New-Keynesian (DNK) models.2 Given its apparent robustness Clarida et
al. (2000), and a large subsequent literature, use the Taylor principle to judge the conduct of monetary policy in practice.
In the present chapter we reassess the usefulness of the Taylor principle. Again, we consider a DNK model featuring firm-specific investment. Sur-prisingly, we find that an active monetary policy is not a sufficient condition for determinacy. This is interesting because most of the existing literature supports the view that the Taylor principle is robust with respect to the modeling of capital accumulation. An exception is Dupor (2001). His result that a passive interest rate rule is required to guarantee determinacy appears, however, to be specific to the continuous time framework he employs. In a discrete-time model Gal´ı et al. (2004) find that it is not endogenous capital
per se that challenges the Taylor principle.3
How is it possible that we reach a different conclusion in the present paper? The answer is that the convenient and widely used assumption of a rental market for capital is not innocuous: it hides an indeterminacy problem. The intuition is as follows. Current investment increases current marginal cost, but it lowers marginal cost in the future. A central bank that follows the Taylor principle therefore tends to decrease future real interest rates in the aftermath of an investment boom. Hence, to the extent that investment
2See, e.g., Taylor (1999) and Woodford (2001).
3Lubik (2003) obtains a similar result. He finds that determinacy obtains under an
is forward-looking, the expectation of such a boom could potentially become self-fulfilling. Whether this possibility materializes, or not, depends on the degree of price stickiness. With sufficiently high price stickiness REE is indeterminate, as we will discuss. The last aspect is crucial for the fact that the rental market assumption hides an indeterminacy problem. As we show in Sveen and Weinke (2003, 2004b) the difference between a specification with firm-specific capital and an alternative formulation with a rental market boils down to a difference in implied price stickiness:4 for any given exogenous
restriction on price adjustment there is less price stickiness, if a rental market for capital is assumed.5 Importantly, with a rental market for capital the
resulting price stickiness will generally be too low to make the indeterminacy issue appear to be relevant from a practical point of view.6 This conclusion
changes if capital is assumed to be firm-specific: if a central bank respects the Taylor principle and follows a rule according to which the nominal interest rate is set as a function of inflation only, then indeterminacy appears to be the regular case.
Based on our results we reinterpret the conduct of monetary policy under Volcker and Greenspan. The analyzes in Clarida et al. (2000) and Lubik and Schorfheide (2004) suggest that the estimated change from a passive to an ac-tive monetary policy explainsin itself the observed stabilization of economic outcomes. We amend their interpretation with a caveat: active monetary policy appears to guarantee desirable macroeconomic outcomes only if it is
4The intuition is analog to the one that explains the difference in implied inflation
dynamics resulting from assuming either constant returns to scale or decreasing returns to scale in a DNK model, along the lines discussed in Sbordone (2002) and Gal´ı et al. (2001).
5The difference in implied price stickiness is therefore a useful metric: Sveen and Weinke
(2004b) show that, for a standard calibration of the two models, one needs a Calvo para-meter of about 0.9 in the rental market model in order to obtain the equilibrium dynamics resulting form a value of 0.75 in the model with firm-specific capital.
6Carlstrom and Fuerst (2003) note that ‘if prices are extremely sticky’ the Taylor
supplemented by interest rate smoothing, and/or some responsiveness of the nominal interest rate to a measure of economic activity. This is precisely the characterization of monetary policy which is empirically plausible under the Volcker-Greenspan tenure.7
The remainder of the chapter is organized as follows: Section 3.2 out-lines the model. Section 3.3 presents our results. In particular, we answer the following three questions. Why is the Taylor principle not sufficient for determinacy in a model with capital accumulation? Why is price stickiness crucial for the indeterminacy issue? Why do interest rate smoothing and responsiveness of the nominal interest rate to economic activity help guar-anteeing determinacy? Section 3.4 concludes.
3.2
The Model
We reconsider the DNK model with firm-specific capital. There are two differences with respect to the framework developed in the first chapter: First, we now assume that the only source of aggregate uncertainty derives from sunspots according to which economic agents coordinate on a particular equilibrium. Second, monetary policy is specified by assuming an interest rate rule. The remaining log-linearized equilibrium conditions are the same as stated in equations (2.1),(2.2),(2.4),(2.5),(2.6), (2.7). The average real marginal cost and marginal savings in labor costs are again obtained in the same way as analyzed in the first chapter.
7For a comprehensive review of the relevant empirical literature on interest rate rules,
3.3
Results
Our goal is to explore what are desirable features of interest rate rules in the sense that they guarantee determinacy. Importantly, the theoretical frame-work developed so far can be used to explain why some rules are more de-sirable than others, as we will see. Finally, we will show that our results are also useful from a positive point of view. They call for a reinterpretation of the conduct of U.S. monetary policy under Volcker and Greenspan.
3.3.1
A Simple Interest Rate Rule
Consider a simple rule according to which the nominal interest rate is set as a function of current inflation:
it=ρ+τππt. (3.1)
We ask what combinations of values for the inflation response coefficient, τπ,
and the price stickiness parameter, θ, result in a determinate equilibrium. The result is shown in Figure 1 for the model with firm-specific capital: a large range of parameter values that meet the Taylor principle are inconsis-tent with determinacy.8 An inflation response coefficient, τ
π, strictly larger
than one is necessary but not sufficient for determinacy. Next we develop the intuition behind this result.
We focus on the role of capital accumulation for equilibrium dynamics. Let us start by conducting a thought experiment. Suppose a sunspot hits the economy and firms increase their investment spending without any change in the economy’s fundamentals justifying it. Could this investment boom be potentially consistent with equilibrium? The answer is yes and the reason
8There is also a standard indeterminacy region in Figure 1. The latter is associated
is simple. Investment has counteracting effects on the determination of the marginal cost. It increases current marginal cost but it reduces marginal cost in subsequent periods. The resulting inflation dynamics inherit the U-shaped marginal cost pattern. In particular, there will be some period of deflation in the aftermath of the investment boom. To the extent that the central bank follows the Taylor principle, the associated real interest rate will therefore drop in the deflationary period. The latter could potentially result in a drop in the long real interest rate relevant for investment.9 If the drop
is sufficiently large, then it may rationalize the investment boom ex post.10
0.5 0.55 0.6 0.65 0.7 0.75 0.8
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
θ
τ π Determinacy
Indeterminacy
Indeterminacy
Figure 1: A simple interest rate rule.
9The long real rate relevant for investment can be written as: rrlong
t = ρ + Et
P∞
k=0βk(rrt+k−ρ),which is obvious from eqaution (2.4).
10The reason for the word ‘sufficiently’ is that the average marginal savings in labor
